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410 BIOLOGY: A. J. LOTKA Proc. N. A. S. 

McEwen, G. F. and Michael, E. L. The functional relation of one variable to each 
of a number of correlated variables determined by a method of successive approxima- 
tion to group averages; a contribution to statistical methods. Proc. Amer. Acad. 
Arts Set., 55, 1919 (95-133). 

Reed, H. S. and Holland, R. H. The growth rate of an annual plant, Helianthus. 
Proc. Nat. Acad. Set., 5, 1919 (135-144). 

Robertson, T. B. On the nature of the autocatalyst of growth. Arch. Entwichl- 
mech., 37, 1913 (497-508). 

1 Paper No. 64, University of California, Graduate School of Tropical Agriculture 
and Citrus Experiment Station, Riverside, California. 

2 The writer wishes to acknowledge his great indebtedness to Dr. G. F. McEwen 
of the Scripps Institution for Biological Research of the University of California for 
valuable assistance in the mathematical work here reported. 



ANALYTICAL NOTE ON CERTAIN RHYTHMIC RELATIONS IN 

ORGANIC SYSTEMS 

By Alfred J. Lotka 

Brooklyn, N. Y. 
Communicated by R. Pearl, May 20, 1920 

Periodic phenomena play an important r61e in nature, both organic and 
inorganic. 

In chemical reactions rhythmic effects have been observed experi- 
mentally, and have also been shown, by the writer 1 and others, 2 to follow, 
under certain conditions, from the laws of chemical dynamics. 

However, in the cases hitherto considered on the basis of chemical 
dynamics, the oscillations were found to be of the damped kind, and 
therefore, only transitory (unlike certain experimentally observed periodic 
reactions). Furthermore, in a much more general investigation by the 
writer, covering the kinetics not only of chemical but also of biological 
systems, it appeared, from the nature of the solution obtained, improbable 3 
that undamped, permanent oscillations would arise in the absence of 
geometrical, structural causes, .in the very comprehensive class of systems 
considered. For it seemed that the occurrence of such permanent oscilla- 
tions, the occurrence of purely imaginary exponents in the exponential 
series solution presented, would demand peculiar and very specific rela- 
tions between the characteristic constants of the systems undergoing 
transformation; whereas in nature these constants would, presumably, 
stand in random relation. 

It was, therefore, with considerable surprise that the writer, on apply- 
ing his method to certain special cases, found these to lead to undamped, 
and hence indefinitely continued, oscillations. 

As the matter presents several features of interest, and illustrates 
certain methods and principles, it appears worth while to set forth the 
argument and conclusions here. 



Vol. 6, 1920 



BIOLOGY: A. J. LOTKA 



411 



Starting out first from a broad basis, we may consider a system in the 
process of evolution, such a system comprising a variety of species of 
matter Si, S 2 . . . .S n of mass X\, Xi. . . .X n . The species of matter S 
may be defined in any suitable way. Some of them may, for example, be 
biological species of organism, others may be components of the "in- 
organic environment." Or, the species of matter S may be several com- 
ponents of an inorganic system in the course of chemical transformation. 

We may think of the state of the system at an instant of time as being 
defined by statement of the values of X\, X$. . . .X„; of certain para- 
meters Q defining the character of each species (in general, variable with 
time); and of certain other parameters P. The parameters P will, in 
general, define the geometrical constraints of the system, both at the 
boundaries (volume, area, extension in space), and also in its interior 
(structure, topography, geography); they will further define such factors 
as temperature and climatic conditions. 

For a very broad class of cases, including those commonly treated in 
chemical dynamics, but extending far beyond the scope of that branch of 
science, the course of events in such a system will be represented by a 
system of differential equations of the form 

§'-*<*.*. X n ;P,Q)) (1) 

(« = 1, 2. . . .n) ) 

If we restrict ourselves to the consideration of evolution at constant 
P's and Q's we may write briefly 

<¥f = Fi(X u X*. . . .XJ. (2) 

The writer has elsewhere 4 given a somewhat detailed discussion of the 
general case, in which no special assumption is made regarding the form 
of the functions F, that is to say, regarding the mode of physical inter- 
dependence of the several species and their environment. 

We now proceed to consider a simple special case, as follows : 

The system comprises 

1. A species of organism Si, a plant species, say, deriving its nourish- 
ment from a source presented in such large excess that the mass of the 
source may be considered constant during the period of time with which 
we are concerned. 

2. A species S2, for example a herbivorous animal species, feeding on Si. 
In this case we have the following obvious relations 

f Other dead 
Mass of Si or excretory 

destroyed by matter elimi- 
S2 per unit of nated from Si 
time per unit of 

I time J 



Rate 



of 

crease of 
per unit 
time 



m- 
Xi 

of 



Mass of newly 
= formed Si per i — 
unit of time 



(3) 



412 




BIOLOGY 


A. 


J. LOTKA 


Proc. N. A. S. 






Mass of newly 


' 




Rate of in- 




formed S% per 




Mass of S 2 




crease of X% 
per unit of 


= -1 


unit of time 
(derived from 


- 


destroyed or 
eliminated per 


(4) 


time 




Si as food in- 
gested) 




unit of time 




Or, in analytical symbols, 




d -§ = A\X 1 -B 1 X 1 X i -A" 1 X 1 
at 


(5) 


= (A\-A" 1 )X 1 -B 1 X 1 X i 


(6) 


= AiXj — -B1A1A2 


(7) 


= X 1 (A 1 -B 1 X 2 ) 


(8) 


—r- = A 2X1X2 — BiXl 

at 


(9) 






= X,(A,Xi 


-£2) 


(10) 



The coefficients A\, An, Bj, B% are in general functions of X\ and X%. 

The reasons for selecting the form (5), (9) for the analytical formula- 
tion of (3), (4) require perhaps a little explanation. For small changes 
the rate of formation of new material of a given species of organism under 
given conditions is proportional to the existing mass of that species. In 
other words, the growth of living matter is a typically autocatakinetic 5 
process. This term has, therefore, been put in the form A' X\ for the 
species Si. Proportionality does not hold for large changes of X\, X%, 
and this is duly provided for in that A '1 is a function of X\, X%. 

Similarly the mass of matter rejected per unit of time from the species 
Si is proportional to X\, and has been put in the form A'\X\, where A" 
is in general a function of X\, Xt. 

Again, the mass of Si destroyed by S 2 feeding upon it will, for small 
changes, be proportional to Xt and also to X\. This term has, therefore, 
been set down in the form B 1X1X2. Here again the departures from pro- 
portionality are taken care of by the variations of Bi with Xi and Xt, of 
which variables B\ is a function. 

Similar remarks apply to the formulation (9) of (4). If there were no 
waste in the feeding process, and assuming that Si consumes no other 
substance than Si, we would have B\ — A 2 ; but in the more general case 
B\ ± A 2 . Approaching now the analytical treatment of equations (5), 
(9), or their equivalents (8), (10), we note first of all that there are two 
ways of satisfying the condition for equilibrium, namely: 

Xi = X 2 = (11) 

and 



X 1 = 



h 

A 2 ' 



Xi = B x 



(12) 



Vol. 6, 1920 



BIOLOGY: A. J. LOTKA 



413 



We shall return later to the condition (11). 

Condition (12) we will employ to define a new origin. Accordingly we 
introduce into (8), (10) the variables: 

(13) 
(14) 



and obtain 



d%\ 

It 

dx?. 

It 



Xi 


= Xi- 


"A 2 


%2 


= x 2 - 


Ai 
~B X 


= 


012*2 4- 


a m %i%2 


= 


021*1 4- 


0212*1*2 



(15) 



where 



012 = — - 



(16) 
(17) 
(18) 
(19) 



B\B% 

a7 

din = — Bi 

AtAz 
On- -g- 

0212 = ^2 

Note the significant fact that in (15) the linear terms in the dexter 
diagonal are lacking. It is this circumstance which imparts an oscillatory 
•character to the process. 

For, since #12 and 021 are in general functions of *i, x% let us expand 
them by Taylor's theorem and put 

an = po + piXi + P2X2 + (20) 

021 = qo 4- 01*1 4- 92*2 4- (21) 

A general solution of the system of differential equations (15) is then 



*i = Pie Xlt 4- Pte M + Pue 2M + P^e 2M + 



* 2 = Qxe Xlt + Qj* + Q u e 2M 4- Q^e 2M + .... 
where Xi, X2, are the roots of the determinental equation for X 

-X po 



qo —X 



(23) 
(24) 

(25) 



that is to say, 



^ = *><p q (26) 

Now, according to (20), (21) p Q , q are the equilibrium values of 012, 
.021. Hence, if we denote by Ai, Bi the equilibrium values of A h B 2 , i.e., 
those values which correspond to *i = * 2 = 0, then we have, by (16), 
.(18) 

poqo = -A1B2 (27) 

and hence 

X = ±V-AiS2 (28) 



414 BIOLOGY: A. J. LOTKA Proc. N. A. S.. 

Now the coefficient 5 2 is, in the nature of things, a positive number, as 
follows from its definition by (4), (9). 

As regards the coefficient A\, we have two possible alternatives. 

If A i is negative for all values of X h X 2 , then X, as denned by (28), 
would be real; but this inference is nugatory. For B\, like B 2 is, by 
definition (3), (5), an essentially positive quantity, and hence the equilib- 
rium defined by (12) would in this case occur at a negative value of Xi. 
But this is physically impossible, since Xi is a mass. 

By referring to (5), (7) it will be seen that this case, in which A\ is 
negative for all values of X h X% and in which an equilibrium of the type 
defined by (12) is physically impossible, corresponds to a species Si in- 
capable of maintaining itself even in the absence of the tax placed upon 
it by the species S 2 feeding upon it. This is a case of minor interest. 

If, on the contrary (12) can be satisfied by a positive value of A\, so 
that an equilibrium of the type (12) is physically possible, then, evidently, 
by (28), X is a pure imaginary. The solution (23), (24) then takes the 
form of Fourier's series; the process is an undamped oscillation con- 
tinuing indefinitely. 

In this connection, it is interesting to recall a passage in Spencer's 
"First Principles," chapter 22, paragraph 173: 

"Every species of plant and animal is perpetually undergoing a rhyth- 
mical variation in number — now from abundance of food and absence of 
enemies rising above its average, and then by a consequent scarcity of 

food and abundance of enemies being depressed below its average 

amid these oscillations produced by their conflict, lies that average 

number of the species at which its expansive tendency is in equilibrium 
with surrounding repressive tendencies. Nor can it be questioned that 
this balancing of the preservative and destructive forces which we see 
going on in every race must necessarily go on. Since increase of numbers 
cannot but continue until increase of mortality stops it, and decrease 
of numbers cannot but continue until it is either arrested by fertility or 
extinguishes the race entirely." 

A question now arises. Do the curves representing the solution (23) , (24) ■ 
dip below the zero axes of X u Xi? This would mean that one or the other, 
or both, of the species Si, Si would become extinct through the violence 
of the oscillations. 

To answer this question we consider the relation: 

dXi _ Xt{AiX\—Bi) ,__. 

dXi ~ XlJAv^BA) 
which is obtained from (8) and (10) by division. From the periodicity 
of x\, xi (and, therefore, Xi, Xi) it follows that the curve defined in rectan- 
gular coordinates X u Xi by (29) is a closed curve. Furthermore, this 
curve can never cross the X\ axis, for at all points of this axis the first,. 



Vol. 6, 1920 BIOLOGY: A. J. LOTKA 415 

and all the higher derivatives of Xi with regard to X\ vanish, as can be 
seen from (29) directly and by successive differentiations. 

Similarly it can be seen that the curve defined by (29) can never cross 
the Xi axis. 

Hence, if any point on any integral curve of (29) lies within the positive 
quadrant, the whole of that curve lies in that quadrant. Thus the 
oscillations can never exceed the limits of positive values Xi, Xz. 

We conclude, therefore, that under the conditions of the problem as 
here set forth, neither the species Si nor the species S2 can become extinct 
through severity of the oscillations alone. In practice the eventuality 
might arise, however, that in the course of these oscillations one or the 
other species might be so thinned out as to succumb to any extraneous 
influence that might arise such as has not been taken into account in our 
present considerations. 

We return now briefly to the consideration of the equilibrium defined 
by the equation 

Xt = X 2 = (11) 

Applying here the criterion set forth by the author elsewhere, 6 it is seen 
that when Ai is positive the determinental equation for X has at this point 
two real roots of opposite sign, which is characteristic of unstable equilib- 
rium. If, on the other hand, Ai is negative in the neighborhood of the 
origin of X lt X% then the equilibrium here is found to be stable, the two 
roots for X being both negative. 

In conclusion it may be remarked that a system of equations identical 
in form with (8), (10) is obtained in the discussion of certain consecutive 
autocatalytic chemical reactions. Here, however, the coefficients A, B 
are constants and the integration can be reduced to a quadrature. Aside 
from a certain number of periodic reactions which have been observed 
more or less as laboratory curiosities, a certain interest is also attached 
to this matter from the fact that rhythmical reactions (e.g., heartbeat, 
which may continue after excision), play an important rdle in physiology. 
We cannot, of course, say whether in such case geometrical (structural) 
features are the dominating factors. 

1 Lotka, A. J., J. Phys. Chem., 14, 1910 (271-274); Zs. pkysik. Ckem., 72, 1910 (508- 
511); 80, 1912 (159-164) ; Phys. Rev., 24, 1912 (235-238) ; Proc. Amer. Acad. Arts Sci., 
55, 1920 (137-153). 

2 Hirniak, J., Zs. physik. Chem., 75, 1910 (675); compare also Lowry and John, J. 
Chem. Soc, 97, 1910 (2634-2645). 

3 Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 145, footnote 13. 

4 Lotka, A. J., Science Progress, 14, 1920 (406-417); Proc. Amer. Acad., loc. cit. 

5 Ostwald, Wo., ttber die zeitlichen Eigenschaften der Entwickelungsvor gauge, Leipsic, 
1908, p. 36. 

6 Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 144, et seq.